Public Math Seminar Good Day To Math. Course Syllabus. July, Instructor : Sung Min Lee Visiting Lecturer : Won Kyung Lee

Transcription

1 Public Math Seminar Good Day To Math Course Syllabus July, 2015 Instructor : Sung Min Lee Visiting Lecturer : Won Kyung Lee Class Details Location : Room 507, Main Building Korea National University of Arts, Seokgwandong Campus, Hwarangro 32-gil, Seongbukgu, Seoul Class Time : 4 to 6 P.M., every Tuesday and Friday. Seminar Overview The seminar, Good Day To Math is composed of seven Mathematics lectures by Sung Min Lee, a graduate student from Carleton College, and one Physics lecture by a visiting lecturer, Won Kyung Lee, a student from Yonsei University. The main objective of the seminar is to deconstruct the stereotype that Math is a study of computation. The seminar does not require any Mathematical background from audience. Each lecture chooses a different topic from Mathematics, and it introduces an idea, covers a number of propositions, and demonstrates proofs rigorously, yet explaining in easy language and providing many examples. For those who cannot attend, the contents of the seminar is also uploaded in a blog for free. Contact Information ryeedmidstal@gmail.com Blog: 1 of 7

2 Table of Contents 1 1. Infinite Sets and Cardinality (07/07) 2. Prime Numbers (07/10) 3. Group Theory (07/14) 4. Elementary Number Theory (07/17) 5. Combinatorics (07/21) 6. Measure Theory (07/24) 7. Unsolved Problems (07/28) Details of Contents 1. Infinite Sets and Cardinality (7th of July, 2015; Class Size: 20) - Sets of Numbers - Natural Numbers (N) - Integers (Z) - Rational Numbers (Q) - Real Numbers (R) - Basic Set Theories - Definition of Elements and Sets - Notations - Functions - Definition of Functions - Injective, Surjective, and Bijective Functions - Comparison of Cardinalities of Finite Sets with Functions - Countable Infinities - Proof that N is equinumerous to the Set of Even Numbers - Proof that Z is equinumerous to N - Proof that Q is equinumerous to Z - Notation of Countable Infinities; Aleph-Null - Uncountable Infinities - Proof that there is no bijective function from [0,1] to N - Claim 2 that [0,1] is equinumerous to R - Notation of Uncountable Infinities; Continuum - A Brief History of Continuum Hypothesis - Fields Medals and International Congress of Mathematicians (ICM) 1 This section only covers the lectures by Sung Min Lee 2 Claim session does not present the proof. The proofs are either skipped or presented in the book I was writing. 2 of 7

3 - Origin of ICM - Hilbert s Speech at ICM in Hilbert s 23 Problems and The First Problem - A Brief History of Gödel s Incompleteness Theorem - Controversy About Set Theory in the 20th Century - Philosophy of Mathematics - Logicism and Formalism - Constructivism and Intuitionism - Russell s Paradox and Barber s Paradox - Kurt Gödel s Incompleteness Theorem - Definition of Completeness and Soundness of Logic System - Existence of Statements of which Truth Values are Independent of Axiom Sets - Incompleteness of Continuum Hypothesis - Paradox of Hotel Hilbert (In Book 3) 2. Prime Numbers (10th of July, 2015; Class Size: 21) - Introduction to Prime Numbers - Definition of Prime Numbers - History of Prime Number Research - Sieve of Eratosthenes - Demonstration of Sieve of Eratosthenes - Trial Division Algorithm and Its Efficiency - Properties of Prime - Unique Factorization Property - Proof that there are infinitely many prime numbers - Proof that there are infinitely many prime numbers congruent to 3 mod 4 - Dirichlet s Theorem - Green-Tao Theorem - Prime Number Theorem - Preliminaries - Prime Counting Function - Natural Constant - Log Function - Tilde Equivalence - Prime Number Theorem - Graphical Demonstration of π(x) and x/ln x - Statistical Evidence - Zeta Function and Its Euler Product Form - History of Basel Problem and Zeta Function - Inducement of Euler Product from Zeta Function 3 In Book materials are presented only in the book, not in the class. 3 of 7

4 - Open Conjectures (In Book) - Twin Prime Conjectures and Its Progress - Goldbach Conjecture and Weak Goldbach Conjecture - Infinitude of Sophie-Germain Primes - Legendre Conjecture - Second Hardy-Littlewood Conjecture 3. Group Theory (14th of July, 2015; Class Size : 15) - Definition of Group - Identity Element - Inverse Element - Closure - Order of an Element - Abelian Groups - Modular Arithmetic and Coprimeness - Definition of Additive Groups - Definition of Multiplicative Groups - Commutativity - Non-Abelian Groups - Definition of Dihedral Groups - Definition of Symmetry Groups - Non-commutativity of a Dihedral Group and Symmetry Groups - Definition of Asymmetry Groups - Operation Table - Operation Table of an Additive Group, Z/4Z - Operation Table of a Multiplicative Group, (Z/10Z) x - Operation Table of a Dihedral Group, D3 - Operation Table of a Symmetry Group, S3 - Group Homomorphism - Showing that the operation tables of Z/4Z and (Z/10Z) x are similar - Showing that the operation tables of D3 and S3 are similar - Definition of Homomorphism - Graphical Explanation of Homomorphism Between D3 and S3 - Graphical Explanation of Homomorphism Between A4 and a Rotation Group of a Tetrahedron 4. Elementary Number Theory (17th of July, 2015; Class Size : 17) - Modular Arithmetic - Definition of Mod - Examples of Congruence - Modular Algebra - Examples of Modular Algebras; ax = b (mod n) 4 of 7

5 - Claim that Number of Solutions Vary Depending On Equations - Euclidean Algorithm - Demonstration of Euclidean Algorithm - Comparison of Its Efficiency to Elementary Method 4 - Claim that gcd(a,b) always divides ax+by for diophantine solutions x,y - Proof that ax = b (mod n) has solutions if and only if gcd(a,n) divides b - Euler Totient Function - Definition and Example of Totient Function - Properties of Totient Function - For n > 2, ϕ(n) is always even - For prime p, ϕ(p) = p-1 - Computing ϕ(n) - Fermat s Little Theorem and Euler s Theorem - Statement and Example of Fermat s Little Theorem and Euler s Theorem - Proof with Group Theory 5. Combinatorics (21st of July, 2015; Class Size : 10) - Fibonacci Sequence - Fibonacci s Rabbits - Generation of Fibonacci Sequence - Generating Function - Definition of Generating Functions - Computing n-th Fibonacci Number with Generating Function - Recurrence Relation - Chessboard Problems - Mutilated Chessboard Problem 5 and Solution - Number of Ways to Cover a 1 by n Chessboard With Monominos and Dominos - Basic Graph Theory - History of Graph Theory; Seven Bridges of Konisberg - Definition of Graphs - Eulerian Circuit and Hamiltonian Circuits - Necessary and Sufficient Conditions for Eulerian/Hamiltonian Circuits - Complete Graphs - Ramsey Theory - Definition of Ramsey Theory - Proof that R(3,3) = 6 - Proof that R(1,n) = 1 - Proof that R(2,n) = n 4 This is the algorithm that is to find gcd(a,b), find prime factorization of a and b respectively and compute the product of shared prime factors of 7

6 6. Measure Theory (24th of July, 2015; Class Size : 11) - Coordinate System - 1-Dimensional Coordinate System - 2-Dimensional Coordinate Systems: Rectangular and Polar Coordinates - 3-Dimensional Coordinate Systems: Rectangular, Cylindrical, and Spherical Coordinates - Definition of Dimensions from Coordinate Systems - Measure Theory - Definition of Lebesgue Measures - 1,2,3-Dimensional Lebesgue Measures - k-dimensional Hausdorff Measure - Relationship Between Hausdorff Measure and Hausdorff Dimension 6 - Fractals and Hausdorff Dimension - Cantor Set - Cantor Dust - Sierpinski Triangle - Von Koch Curve 7. Unsolved Problems (28th of July, 2015; Class Size : 11) - Fermat s Last Theorem (FLT) - Who is Pierre de Fermat? - FLT Briefing - Definition of Elliptic Curves - Taniyama-Shimura Conjecture (TSC) - Frey s Equation (FE) - Ribet s Theorem (RT) - Correlation of FLT, TSC, FE, and RT - Who is Andrew Wiles? - Poincare Conjecture (PC) - Who is Henri Poincare? - PC Briefing - Definition of Manifolds, n-sphere, Simply-Connected Space, Compactness, and Homeomorphism - Example of Non-Simply-Connected Space - Example of Homeomorphism; Coffee Cup and Donut - Progress of Generalized Poincare Conjecture; Milnor, Smale, and Freedman - Who is Grigori Perelman? - Four Color Theorem (FCT) 6 Let A be a set, and ka be the set scaled by k. Assuming m(ka) = k^x m(a), then the dimension of A can be defined as x. 6 of 7

7 - FCT Briefing - Origin of FCT - Graph Theoretical Approach - Planar Graph - Proof of Six Color Theorem - Controversy about Haken and Appel s Approach - Riemann Hypothesis (RH) - Who is Bernhard Riemann? - RH Briefing - Definition of Zeta Function, Complex Number, and Real Parts - Analytic Continuation of Zeta Function - Definition of Trivial Zeroes - Graphical Demonstration of Zeta Function on Complex Plane and Its Non-Trivial Zeroes - Rumors and Taboos Related to Riemann Hypothesis - Montgomery and Dyson s Discovery 7 - Will a proof to the RH affect security of RSA? - Possible Scenarios about RH 7 They discovered that an equation of atomic energy level distribution and distribution of nontrivial solutions share a similarity. 7 of 7

8 7 월매주화 / 금, 오후 4 시 -6 시, 총 8 강 한국예술종합학교본부 5 층강의실 507 호